Distribution of Laplacian Eigenvalues of Graphs

Spectral graph theory (Algebraic graph theory) is the study of spectral properties

of matrices associated to graphs. The spectral properties include the study of characteristic

polynomial, eigenvalues and eigenvectors of matrices associated to graphs.

This also includes the graphs associated to algebraic structures like groups, rings

and vector spaces. The major source of research in spectral graph theory has been

the study of relationship between the structural and spectral properties of graphs.

Another source has research in mathematical chemistry (theoretical/quantum chemistry).

One of the major problems in spectral graph theory lies in finding the spectrum

of matrices associated to graphs completely or in terms of spectrum of simpler

matrices associated with the structure of the graph. Another problem which is worth

to mention is to characterise the extremal graphs among all the graphs or among a

special class of graphs with respect to a given graph, like spectral radius, the second

largest eigenvalue, the smallest eigenvalue, the second smallest eigenvalue, the graph

energy and multiplicities of the eigenvalues that can be associated with the graph

matrix. The main aim is to discuss the principal properties and structure of a graph

from its eigenvalues. It has been observed that the eigenvalues of graphs are closely

related to all graph parameters, linking one property to another.

Spectral graph theory has a wide range of applications to other areas of mathematical

science and to other areas of sciences which include Computer Science,

Physics, Chemistry, Biology, Statistics, Engineering etc. The study of graph eigen- values has rich connections with many other areas of mathematics. An important development is the interaction between spectral graph theory and differential geometry. There is an interesting connection between spectral Riemannian geometry

and spectral graph theory. Graph operations help in partitioning of the embedding

space, maximising inter-cluster affinity and minimising inter-cluster proximity. Spectral

graph theory plays a major role in deforming the embedding spaces in geometry.

Graph spectra helps us in making conclusions that we cannot recognize the shapes

of solids by their sounds. Algebraic spectral methods are also useful in studying the

groups and the rings in a new light. This new developing field investigates the spectrum

of graphs associated with the algebraic structures like groups and rings. The

main motive to study these algebraic structures graphically using spectral analysis

is to explore several properties of interest.